For learning about the auxiliary variables for the parent prior parameters, we use the expectations from Eq. REF  and Eq. REF  which now can be directly calculated given category counts. We do not have to explicitly calculate variational distributions over the new auxiliary variables \(n^{\prime }\)  and \(m^{\prime }\) , which allows for an efficient inference.
If we separate our parent prior parameters into a mean and a precision part as in [1]}, the multinomial mean part of the parent parameters follow a Dirichlet distribution, while the precision part follows a Gamma function, as shown in [2]}.
Using the auxiliary variable scheme, it is also possible to integrate over Dirichlet distributed means of parent priors, which allows for collapsed inference schemes for hierarchical multi-Dirichlet and multi-Dirichlet process models.
